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jmc

counting and probability senior

Problem

A pirate is searching for buried treasure on 6 islands. On each island, there is a chance that the island has buried treasure and no traps, a chance that the island has traps but no treasure, and a chance that the island has neither traps nor treasure. What is the probability that while searching all 6 islands, the pirate will encounter exactly 3 islands with treasure, and none with traps?
Solution
There are ways to choose 3 islands. For each of these choices, there is a probability of that the chosen islands have treasure and the remaining ones have neither treasure nor traps. Therefore, the probability that the pirate encounters exactly 3 islands with treasure and none with traps is .
Final answer
\frac{5}{54}